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Let $\Sigma$ be the surface in $\mathbb{R}^3$ parametrized by $$ (u,v) \mapsto \Big(\;p_X(u,v),\; p_Y(u,v),\; p_Z(u,v)\;\Big), $$ where $p_X, p_Y, p_Z$ are polynomials. Is there a standard way to obtain the implicit description of $\Sigma$? I mean, to find polynomials $f$ and $g$ in $X,Y,Z$ such that $\Sigma$ is the zero locus of the ideal generated by $f$ and $g$?

Even if you don't know an algorithm to do that, do you know if at least it is alway possible (theoretically) to find such polynomials $f,g$ ?

Abramo
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2 Answers2

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If your computing environment supports Gröbner basis methods, then you can use that to retrieve the implicit Cartesian equation of your parametrically defined surface.

Mathematica, for instance, has the function GroebnerBasis[]. As an example, here is how you might use it to derive the implicit Cartesian equation for the Enneper minimal surface:

GroebnerBasis[{x, y, z} == {u - u^3/3 + u v^2, -v - u^2 v + v^3/3,
                            u^2 - v^2} // Thread,
              {x, y, z}, {u, v}] // FullSimplify
   {729 x^6 - (9 y^2 - 8 (-3 + z) z^2)^2 (9 y^2 + z (3 + z)^2) +
    27 x^2 (81 y^4 + 18 y^2 z (3 + 13 z^2) +
    16 z^3 (3 + z) (-3 + z (6 + z))) -
    243 x^4 (9 y^2 + z (3 - z (18 + 5 z)))}
J. M. ain't a mathematician
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I found this article, where they show that an implicit equation is always there for parametrisations of degree $\leq3$. The technique uses the so called called $\mu$-basis

http://staff.ustc.edu.cn/dengjs/files/papers/47%20mu23.pdf

Abramo
  • 6,917